Algorithmic aspects of Roman domination in graphs

Abstract

For a simple, undirected graph $$G = (V, E)$$ , a Roman dominating function (RDF) $$f{:}V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that, every vertex u with $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) = 2$$ . The weight of a RDF is the sum $$f(V) = \sum _{v \in V}f(v)$$ . The minimum weight of a RDF is called the Roman domination number and is denoted by $$\gamma _{R}(G)$$ . Given a graph G and a positive integer k, the Roman domination problem (RDP) is to check whether G has a RDF of weight at most k. The RDP is known to be NP-complete for bipartite graphs. We strengthen this result by showing that this problem remains NP-complete for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. We show that $$\gamma _{R}(G)$$ is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph. We show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within $$(1 - \epsilon ) \ln |V|$$ for any $$\epsilon > 0$$ unless $$NP \subseteq DTIME(|V|^{O(\log \log |V|)})$$ and also propose a $$2(1+\ln (\varDelta +1))$$ -approximation algorithm for the MRDP, where $$\varDelta $$ is the maximum degree of G. Finally, we show that the MRDP is APX-complete for graphs with maximum degree 5.